 Phase mixing in magnetic flux expulsion

Differential circulation of a weak magnetic field winds its flux function into progressively smaller radial scales. In a smooth eddy with angular-velocity gradient of order $U/L^2$, its radial wavenumber grows as $k_r\sim Ut/L^2$. Resistive damping becomes substantial when $\eta {\int }_0^t{k}_r^{2}dt\sim 1$, giving $t\sim (L^4/(\eta U^2){)}^{1/3}=(L/U){\mathrm{Rm}}^{1/3}$. This estimate presumes differential rotation and negligible magnetic feedback; a rigidly rotating eddy does not create this phase mixing. Finite magnetic diffusivity then permits weak interior field and boundary concentration, unlike exact ideal flux freezing.